Here, \(k\)  specifies a component of the environment \(\mathcal {E}\)  of size
\({^{\sharp }}\hspace{-1.42262pt}\mathcal {E}\) . The operators, \(\Upsilon _{k}\)  and \(\Omega _{k}\) , are arbitrary. This
class of models contains the c-maybe model of Touil et al. [1]}.
That model has \(\hat{\Pi }_{\mathcal {S}} =0\cdot |0\rangle \langle 0|+1\cdot |1\rangle \langle 1|\)  and \(\exp \left[\imath \Upsilon _{k}t\right]=\sin a|0\rangle \langle 0|+\cos a\left(|0\rangle \langle 1|+|1\rangle \langle 0|\right)-\sin a|1\rangle \langle 1|\) 
for all \(k\) , where \(a\)  is the angle of rotation of the “target”
environment bit after a time \(t\)  Note that all the coupling frequencies (i.e., the energy scales divided
by the reduced Planck's constant) are absorbed into the definition
of the operators \(\mathbf {H}_{\mathcal {S}}\) , \(\Upsilon _{k}\)  , and \(\Omega _{k}\) , while \(\hat{\Pi }_{\mathcal {S}} \)  is
dimensionless.. All other operators are 0. The collection of operators act similarly
to those in the controlled NOT gate. They just swap too, only a bit
more lazy, as here \(a\) 's any number, so it's called c-maybe.
